The relationships of these various maps to one another are illustrated in the following commutative diagram.

The equivariance condition can also be understood as the following commutative diagram.

The composition of morphisms is often represented by a commutative diagram.

The situation is described by the following commutative diagram:

Such a sequence is shorthand for the following commutative diagram:

The associated commutative diagram is a morphism of fiber bundles.

These statements are equivalent in that they are expressed by the same commutative diagrams.

We can rewrite these conditions using following commutative diagrams:

Conversely, given a commutative diagram, it defines a poset category:

Then we obtain a commutative diagram in which all the diagonals are short exact sequences:

That is to say, the adjunction space is universal with respect to following commutative diagram:

Commutative diagrams play the role in category theory that equations play in algebra.

That is, the following diagram Commutative diagram (the horizontal maps are the usual ones):

For clarification, phrases like "this commutative diagram" or "the diagram commutes" may be used.

This means that, formulated as commutative diagrams, the usual associativity and inversion axioms of a group continue to hold.

The snake lemma shows how a commutative diagram with two exact rows gives rise to a longer exact sequence.

Manipulation and visualization of objects, morphisms, commutative diagrams, categories, functors, natural transformations.

The usual functorial axioms are replaced with corresponding commutative diagrams involving these morphisms.

We sketch the proof for the non-trivial part of the theorem, corresponding to the sequence of commutative diagrams on the right.

Since X is separated over S and is dense, this is clear from looking at the relevant commutative diagram.

In a monoidal category, analogs of usual monoids from abstract algebra can be defined using the same commutative diagrams.

The axioms of unital associative algebras can be formulated in terms of commutative diagrams.

Conversely, every commutative diagram represents a diagram (a functor from a poset index category) in this way.

More formally, a commutative diagram is a visualization of a diagram indexed by a poset category:

See the article on natural transformations for the explanation of the notations and , or see below the commutative diagrams not using these notions: